Properties

Label 2-588-1.1-c1-0-1
Degree $2$
Conductor $588$
Sign $1$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s + 9-s + 2·11-s + 6·13-s − 4·15-s + 4·17-s + 4·19-s + 2·23-s + 11·25-s + 27-s − 2·29-s + 2·33-s + 2·37-s + 6·39-s − 4·43-s − 4·45-s − 12·47-s + 4·51-s − 6·53-s − 8·55-s + 4·57-s + 8·59-s − 6·61-s − 24·65-s − 8·67-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.417·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 0.348·33-s + 0.328·37-s + 0.960·39-s − 0.609·43-s − 0.596·45-s − 1.75·47-s + 0.560·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.04·59-s − 0.768·61-s − 2.97·65-s − 0.977·67-s + 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486002552\)
\(L(\frac12)\) \(\approx\) \(1.486002552\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95477629177032576079496808423, −9.694844863101715383876080158734, −8.708229597668417567477010531317, −8.077019225803717160884657215512, −7.38098041713554163025681861500, −6.34426698354997241941479748412, −4.87924854330324804890540113594, −3.65675458100353588864150742280, −3.37164547236104444198213794231, −1.14653761218391715748516515519, 1.14653761218391715748516515519, 3.37164547236104444198213794231, 3.65675458100353588864150742280, 4.87924854330324804890540113594, 6.34426698354997241941479748412, 7.38098041713554163025681861500, 8.077019225803717160884657215512, 8.708229597668417567477010531317, 9.694844863101715383876080158734, 10.95477629177032576079496808423

Graph of the $Z$-function along the critical line